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L11n43 L11n45

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X5,14,6,15 X3849 X9,18,10,19 X11,20,12,21 X13,22,14,5 X19,10,20,11 X21,12,22,13 X15,2,16,3

Gauss Code:

{{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 9, -7, 10, -8, 4, -11, -2, 3, 6, -9, 7, -10, 8}}

Jones Polynomial:

q-27/2 - q-17/2 - q-13/2 - q-9/2

A2 (sl(3)) Invariant:

- q-44 - q-42 - 2q-40 - 2q-38 - 2q-36 - q-34 + q-32 + 2q-30 + 3q-28 + 3q-26 + 3q-24 + 2q-22 + 2q-20 + q-18 + q-16

HOMFLY-PT Polynomial:

- 4a9z-1 - 19a9z - 36a9z3 - 28a9z5 - 9a9z7 - a9z9 + 7a11z-1 + 21a11z + 21a11z3 + 8a11z5 + a11z7 - 3a13z-1 - 4a13z - a13z3

Kauffman Polynomial:

4a9z-1 - 19a9z + 36a9z3 - 28a9z5 + 9a9z7 - a9z9 - 7a10 + 21a10z2 - 21a10z4 + 8a10z6 - a10z8 + 7a11z-1 - 28a11z + 42a11z3 - 29a11z5 + 9a11z7 - a11z9 - 7a12 + 21a12z2 - 21a12z4 + 8a12z6 - a12z8 + 3a13z-1 - 9a13z + 6a13z3 - a13z5 - a18

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -8                   1 j = -10                   1 j = -12               1     j = -14           1         j = -16           2 1       j = -18       1             j = -20       1 1           j = -22   1 1               j = -24   1 1               j = -26 1                   j = -28 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 44]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 44]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 14, 6, 15], > X[3, 8, 4, 9], X[9, 18, 10, 19], X[11, 20, 12, 21], X[13, 22, 14, 5], > X[19, 10, 20, 11], X[21, 12, 22, 13], X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 9, -7, 10, -8, 4, -11, -2, 3, 6, > -9, 7, -10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) -(17/2) -(13/2) -(9/2) q - q - q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-44 -42 2 2 2 -34 -32 2 3 3 3 2 -q - q - --- - --- - --- - q + q + --- + --- + --- + --- + --- + 40 38 36 30 28 26 24 22 q q q q q q q q 2 -18 -16 > --- + q + q 20 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 44]][a, z]
Out[8]=	9 11 13 -4 a 7 a 3 a 9 11 13 9 3 11 3 ----- + ----- - ----- - 19 a z + 21 a z - 4 a z - 36 a z + 21 a z - z z z 13 3 9 5 11 5 9 7 11 7 9 9 > a z - 28 a z + 8 a z - 9 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 44]][a, z]
Out[9]=	9 11 13 10 12 18 4 a 7 a 3 a 9 11 13 -7 a - 7 a - a + ---- + ----- + ----- - 19 a z - 28 a z - 9 a z + z z z 10 2 12 2 9 3 11 3 13 3 10 4 > 21 a z + 21 a z + 36 a z + 42 a z + 6 a z - 21 a z - 12 4 9 5 11 5 13 5 10 6 12 6 9 7 > 21 a z - 28 a z - 29 a z - a z + 8 a z + 8 a z + 9 a z + 11 7 10 8 12 8 9 9 11 9 > 9 a z - a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-10 -8 1 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 28 9 26 9 24 8 22 8 24 7 22 7 20 6 q t q t q t q t q t q t q t 1 1 2 1 1 1 > ------ + ------ + ------ + ------ + ------ + ------ 18 6 20 5 16 4 14 4 16 3 12 2 q t q t q t q t q t q t

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